DOI 10.1140/epjc/s10052-009-1176-y Regular Article - Theoretical Physics
Scalar-tensor theory and the anisotropic perturbations
of the inflationary universe
W.F. Kaoa
Institute of Physics, Chiao Tung University, Hsinchu, Taiwan
Received: 17 September 2008 / Revised: 26 November 2008 / Published online: 31 October 2009 © Springer-Verlag / Società Italiana di Fisica 2009
Abstract Inflationary higher derivative scalar-tensor the-ory is analyzed in this paper in a de Sitter background space. A useful model-independent formula of the Fried-mann equation is derived and used to study the stability problem associated with the anisotropic perturbations of the inflationary solution. The stability conditions of the de Sit-ter solution are derived for a general class of models. For a simple demonstration, an induced gravity model is con-sidered in this paper for the effects of the higher derivative interactions including a cubic term.
PACS 98.80.Cq· 04.20.-q · 04.20.Cv
1 Introduction
The physical universe is a highly homogeneous and isotropic [1, 2] space known as the Friedmann–Robertson–Walker (FRW) space [3–6]. The cosmological problems, such as the flatness, the monopole, and the horizon problem, associ-ated with the standard big bang model can be resolved by a successful inflationary mechanism [7–10].
Moreover, the Einstein–Hilbert models are expected to acquire higher derivative modifications near the Planck scale [11,12]. For example, the quantum gravity and the string theories both show that the higher derivative terms could have interesting cosmological implications [11,12] in the high energy domain. On the other hand, the higher deriva-tive terms can also be interpreted as the quantum corrections of the matter fields [13–15]. Therefore, the possibility of de-riving inflation from the higher derivative corrections has been a focus of research interest for a long time [14–18]. In addition, a general analysis on the stability conditions of the gravity theories is also useful in the search of the compatible physical models with our physical universe. For example,
ae-mail:gore@mail.nctu.edu.tw
the stability conditions for a variety of pure gravity theories as a potential candidate of inflationary universe in the flat Friedmann–Robertson–Walker (FRW) space has been dis-cussed in detail in Refs. [16,17,19–24].
It is known that any stable isotropically expanding solu-tion should also be stable against any anisotropic perturba-tion. In fact, our physical universe could be anisotropic in the early stage of the evolution. It is therefore interesting to study the stability conditions derived from the anisotropic perturbations against a de Sitter expanding space during the early epoch. For instance, it has been shown that an infla-tionary solution does exist for an NS-NS model with a met-ric, a dilaton, and an axion field [25] in a Bianchi space. This inflationary solution can be shown to be stable against small anisotropic perturbations [26]. Similar analysis has also been studied for a variety of models [27].
Recently, there are growing interests in the study of the Kantowski–Sachs (KS) type spaces [28–30]. We will hence propose to study the existence and stability problem of the inflationary solution in a KS space. In particular, we will focus on the effects of the higher derivative terms in the KS space. Note that the stability analysis for a large class of pure gravity models admitting an inflationary KS/FRW solution was presented in Refs. [31,32]. It is shown that the stability conditions of the de Sitter background space are closely re-lated to the choice of the coupling constants in these models. For later convenience, any KS type solution that approaches asymptotically to a FRW final de Sitter state will be referred to as the KS/FRW solution in this paper.
The perturbation equations for any small anisotropic per-turbations in a KS type space are identical to the perturba-tion equaperturba-tions for any small isotropic perturbaperturba-tions against the isotropic de Sitter space in the inflationary phase [16, 17,19,20]. Therefore, the existence of an unstable mode of the perturbation equations will ensure a proper resolution for the graceful exit problem both for the anisotropic per-turbations and the isotropic perper-turbations in the inflationary phase as long as they both approach the same final de Sitter
state. In certain models, unstable modes may not exist for the pure gravity models. A slow roll-over scalar field could hopefully provide a possible alternative to this problem. The slow roll-over scalar field will hold the de Sitter phase stable for a brief moment before the inflationary phase comes to an end. The anisotropic perturbations of these models may, however, not be compatible with the slow changing scalar field. Therefore, we need to solve the perturbation equations carefully in order to find out possible constraints on the cou-pling constants in these models. Once the inflation is over, the stable modes will ensure that the de Sitter space can remain stable and anisotropy will not grow out of control. Therefore, the absence of an unstable mode in the post in-flationary era is also critical to the stability of the de Sitter background.
To be more specifically, an inflationary de Sitter solution in a scalar-tensor theory must have at least an unstable mode for the perturbation in δφ or some linear combinations of the
δHand δφ. Accordingly, the inflationary era will come to an end once the unstable mode takes over after a brief period
t of inflationary expansion. If this period t is not long enough to derive 60-e fold inflationary expansion, the infla-tionary phase will be ended and wipe out the effect of the slow-rolling scalar field. Therefore, a unstable mode with a reasonable large unstable period t is needed for the grace-ful exit problem. We will show that the scalar field does pro-vide a proper resolution to the graceful exit problem in this paper for in the higher derivative induced gravity models.
This paper will be organized as follows: (1) the derivation of a simple and model-independent formula of the Fried-mann equation for a pure gravity theory will be reviewed briefly in Sect.2; (2) a more general and model-independent stability analysis of the higher derivative scalar tensor mod-els will be presented in Sect.3; (3) in Sect.4, we will fo-cus on the higher derivative induced model with a cubic La-grangian as an explicit example; (4) the conclusions will be presented in Sect.5.
2 The Friedmann equation and the Bianchi identity in a KS space
The metric of the Kantowski–Sachs type space can be writ-ten as
ds2= −dt2+ c2(t ) dr2+ a2(t )d2θ+ f2(θ ) dϕ2 (1) with f (θ )= (θ, sinh θ, sin θ) denoting the flat, open and close anisotropic space. More specifically, the Bianchi I (BI), III (BIII), and Kantowski–Sachs (KS) space corre-sponds to the flat, open and closed model respectively. This metric can also be written as
ds2= −dt2+ a2(t ) dr2 1− kr2+ r 2dθ2 + a2 z(t ) dz2 (2)
with r and θ the polar coordinates, and z as the z-coordinate. Note that k= 0, 1, −1 stands for the flat, open and closed universes similar to the FRW space when a= az.
Writing Hμν≡ Gμν− Tμν, the Einstein equation can be written as DμHμν= 0 by incorporating the Bianchi iden-tity, DμGμν= 0, and the energy momentum conservation,
DμTμν= 0. Here Gμν and Tμνrepresent the Einstein ten-sor and the energy momentum tenten-sor coupled to the sys-tem respectively. With the metric (2), it can be shown that the r component of the equation DμHμν= 0 implies that
Hrr = Hθθ.This result also says that any matter coupled to the system must have the symmetric property Trr = Tθθ. In addition, the equations DμHμθ = 0 and DμHμz= 0 both vanish identically for all kinds of energy momentum ten-sors. The most interesting information comes from the t component of this equation. It says that (∂t + 3H )Htt = 2H1Hrr+ HzHzz.This equation asserts that (i) Htt = 0 im-plies that Hrr = Hzz= 0 and (ii) Hrr = Hzz= 0 only implies
(∂t + 3H )Htt = 0 instead of Htt = 0. The case (ii) can be solved to give Ht
t = constant × exp[−a2az] that approaches zero when a2az→ ∞.
For an anisotropic KS space, the metric contains two independent variables a and az. The Einstein field equa-tions have, however, three non-vanishing components, i.e.,
Htt = 0, Hrr = Hθθ = 0 and Hzz= 0. The Bianchi identity
implies that the tt component is not redundant and hence must be retained for a complete analysis. Ignoring either one of the rr or zz components will not, however, affect the final result of the system. In short, the Htt = 0 equation, known as the generalized Friedmann equation, is a non-redundant field equation as compared to the Hrr= 0 and Hzz= 0 equa-tions.
By restoring the gt t component b2(t )= 1/B1 will be helpful in deriving the non-redundant field equation associ-ated with Gt t that will be shown shortly. More specifically, we will introduce the generalized KS metric:
ds2= −b2(t ) dt2+ a2(t ) dr2 1− kr2 + r 2dθ2 + a2 z(t ) dz2 (3)
for the following reasons. In principle, the Lagrangian of the system should reduce from a functional of the metric gμν, or equivalently L(gμν), to a simpler function of a(t) and
az(t ), namely L(a(t), az(t )) ≡ a2azL(gμν(a(t ), az(t ))). The equation of motion as a function of a(t) and az(t ) should be derivable from the variation of the effective La-grangian L(t) with respect to the variable a and az. The result is, however, incomplete because the variation of a and az are related to the variation of grr and gzz respec-tively. The field equation from the variation of gt tcannot be derived from the effective Lagrangian without restoring the variable b(t) in advance. This is the motivation to introduce
the metric (3) such that the effective Lagrangian L(t)≡
ba2azL(gμν(b(t ), a(t ), az(t ))) restores the non-redundant information hidden in the Htt = 0 equation associated with the variation of the gt t equation. The non-redundant Fried-mann equation can hence be reproduced by setting b= 1 after the variation of b(t) has been done.
Note that all non-vanishing components of the curvature tensor can be computed as [31,32]
Rtjt i= 1 2 ˙B1Hi+ B1 ˙Hi+ H 2 i δji (4) Rijkl= B1HiHj ij m klm+ k a2 ij z klz (5)
with Hi≡ (˙a/a, ˙a/a, ˙az/az)≡ (H1, H2= H1, Hz)for r, θ , and z component respectively.
Given a pure gravity model with a reduced Lagrangian
L= √gL = L(b(t), a((t), az(t )), it can be shown that
L=a 2a z √ B1 LRtjt i, Rklij=a 2a z √ B1 LHi, ˙Hi, a2 (6)
with B1= b−2 for convenience. As mentioned earlier the
Friedmann equation can be derived from the variational equation with respect to the δB1(= δb−2 = −2δb/b3)
-equation of the reduced Lagrangian L. Our task here is to replace all δB1and δ ˙B1effectively with δHi and δ ˙Hi such that before we can set B1= 1 freely without any trouble and
write the Friedmann equation free of the function b(t). As a result, we can derive the field equations directly from the gij components more easily without bothering the restoration of the gt t information any more. As a result, the Friedmann equation can be obtained from the above method by replac-ing δL/δB1and δL/δ ˙B1with some proper combinations of
δL/δHi and δL/δ ˙Hi.
As a result, the Friedmann equation for the pure gravity model L can be shown to be [31,32]
DL≡ L + Hi d dt + 3H Li− HiLi− ˙HiLi= 0 (7) DzL≡ L + d dt + 3H 2 Lz− d dt + 3H Lz= 0 (8) Here Li ≡ δL/δHi, Li ≡ δL/δ ˙Hi, and 3H ≡ iHi. The second equation is derived from the variation equation δaz. For simplicity, we have written L as L in the above equa-tions. Note again that the δa1equation is redundant
follow-ing the Bianchi identity shown above.
The proof follows from an observation that ˙B1 always
shows up as a combination of ˙B1Hi+2B1( ˙Hi+Hi2). There-fore δL/δ ˙B1= HiδL/[2δ ˙Hi]. Here we have set B1= 1 whenever it will not affect the final result. Moreover, the summation over repeated indices is not written explicitly. In addition, δL/δB1= HiδL/[2δHi] + ˙HiδL/δ ˙Hi if L=
L(B1(aiH˙i+ aijHiHj))for any arbitrary “constant” coef-ficients ai and aij. In fact, it can be shown that this result holds for all anisotropic Bianchi type spaces including the KS type spaces shown in (4)–(5). Indeed, the term B1H˙iwill always show up together with B1HiHj from the dimension analysis. Therefore the Friedmann equation derived above is a universal formula holds for all homogeneous Bianchi type spaces.
3 Higher derivative scalar tensor model
With an additional scalar field Lagrangian Lφcoupled to the scalar tensor Lagrangian Lg, we will have
L= a fa(φ)L(a)+ Lφ ≡ a fa(φ)L(a)(Hi, ˙Hi)− 1 2∂μφ∂ μφ− V (φ) (9)
with fa(φ)some polynomial functions of φ and L(a)some
ath order pure gravity Lagrangian. These models are also known as modified gravity theories. For example, f1(φ)=
φ2/2, L(1) = −R, f2= −α, L(2)= R2, f3(φ)= γ φ−2
and L(3)= R3 stand for the induced gravity model of the
Einstein–Hilbert action, the quadratic term and the cubic La-grangian of the system. Here V (φ) denotes the scalar field potential coupled to the gravitational system.
The Friedmann equation becomes
D i fa(φ)L(a) = a fa(φ)DL(a)+ a Hifa(φ) ˙φ Li(a) =1 2 ˙φ 2+ V (φ) (10)
for this model. In addition, the scalar field equation can be shown to be
¨φ + 3H0˙φ + V=
a
fa(φ)L(a) (11)
We will focus on the stability problem of an inflationary de Sitter background solution Hi = H0 and φ= φ0 with a constant Hubble parameter H0and a constant initial scalar
field φ0. Let Hi = H0+ δHi and φ= φ0 + δφ be the anisotropic perturbations against the constant de Sitter back-ground space. As a result, we have
a fa(φ0)DL(a)(Hi= H0)= V (φ0) (12) V(φ0)= a fa(φ0)L(a) (13)
as the leading zeroth-order equations of the perturbation equations.
The first-order perturbation equations of the pure gravity part of DL can be shown to be
δ(DL)=HiLijδ ¨Hj + 3HHiLijδ ˙Hj + 3HHiLij+ L jδH j +HiLi δ(3H ) − HiLijδHj (14)
for any DL(a)defined by (7) with all functions of Hi eval-uated in the de Sitter background Hi = H0. From now on, the notationAiBi ≡
i=1,zAiBi denotes the summation over i= 1 and z for repeated indices. Note that we have ab-sorbed the information of i= 2 into i = 1. They contributes equally to the field equations in the KS type spaces. In addi-tion, Lij ≡ δ2L(a)/δ ˙HiδHj and similarly for Lij and Lij. Here the upper index i and the lower index j denote the variation with respect to ˙Hi and Hj respectively for conve-nience. Note that the perturbation equation associated with (8) can also be shown to be the same as (14) in the de Sitter space due to the symmetry of the de Sitter background space [7–10].
In addition, it can be shown that HiLi1 = 2HiLiz, HiLi1 = 2HiLiz, L1= 2Lz, HiLi1 = 2HiLiz, and
L1= 2Lz for a KS type space approaching the inflationary de Sitter background metric with Hi= H0. As a result, the stability equations (14) can be greatly simplified. For conve-nience, we will also define the operatorDLas
DLδH ≡ HiLi1 δ ¨H+ 3HHiLi1 δ ˙H + 3HHiLi1+ L1 δH+ 2HiLi δH − HiLi1δH (15)
This equation hence becomes
DLδH = H0 Li1 δ ¨H+ 3H0Li1 δ ˙H +3H0Li1+ L 1 + 2Li − L i1 δH (16) = H0Li1 δ ¨H+ 3H0δ ˙H (17) + 3H0Li1+ L1 + 2Li − Li1 Li1 δH
when the constant Hubble parameter is written explicitly. For convenience, we will also writeDL(a)δH= DaδH. As
a result, the stability equation can be written as
δ(DL(a))= Da δH1+ δHz 2 =3 2Da(δH ) (18) with H= (2H1+ Hz)/3 as the mean value of all Hi.
Hence the first-order perturbation equation in δH and δφ of the Friedmann equation can be shown to be
3 2 a fa(φ0)DaδH= V(φ0)− a fa(φ0)DL(a) δφ −3 2 a H0fa(φ0) L1(a)δ ˙φ (19)
Therefore, we will be solving the following equation: 3 2H0 Li1 δ ¨H+ 3H0δ ˙H+ KH02δH = V(φ0)− a fa(φ0)DL(a) δφ −3 2 a H0fa(φ0) L1(a)δ ˙φ (20) with K≡3H0L i 1+ L1 + 2Li − Li1 Li1H2 0 ,
and L=afa(φ0)L(a)the total coefficient K and the to-tal gravitational Lagrangian respectively in the constant φ0
and H0background space. The explicit expression of K
de-pends on the models being considered. The values of K plays, however, some crucial rolls in the stability problem of the corresponding de Sitter universe. Some general selec-tion rules can be obtained in a straightforward way. Simi-larly, the first order perturbation equation of the scalar field can be shown to be δ ¨φ+ 3H0δ ˙φ+ J H02δφ =3 2 a fa(φ0) L1(a)δ ˙H+ L(a)1δH (21) with J= V0− a fa(φ0)L(a) H0−2
In addition, the variational equation of δaz can be shown explicitly to be redundant in the limit Hi = H0+ δHi and
φ= φ0+ δφ following the Bianchi identity. In summary,
the values of J and K will affect the stability of the de Sit-ter solution. The above equations hence provide a model-independent method in determining whether a model is compatible with the inflationary de Sitter universe.
Indeed, by assuming that δH= exp[hH0t]δH0and δφ= exp[pH0t]δφ0for some constants h and p, we can write the above equations as 3 2H 3 0 Li1 h2+ 3h + K δH
= −3 2 a H02fa(φ0)L1(a)[p − J1]δφ (22) H02p2+ 3p + Jδφ =3 2 a fa(φ0)L1(a)H0[h + K1]δH (23) with K1= [afa(φ0)L(a)1]/[ afa(φ0)L1(a)H0] and J1= 2 V(φ0)− afa(φ0)DL(a) 3aH02fa(φ0)L1(a)
Note that the perturbation equation of δH shown on the left-hand side of (22) is the same as the pure gravity model with similar coupling constants. This equation is also the same as the perturbation equation in their isotropic limit.
We will show that the stability of the anisotropic space depends on the coefficient K. Indeed, (22) and (23) indicate that there are two decaying modes for δH and δφ with
2h= −3 ±√9− 4K = −2K1 (24)
2p= −3 ±√9− 4J = 2J1 (25)
As a result the modified gravity models are subjected to strong constraints in order to accommodate a consistent per-turbative de Sitter inflationary solution:
Hi= H0+ Aiexp[−K1H0t] (26)
φ= φ0+ δφ0exp[J1H0t] (27)
Here Ai and δφ0are small initial perturbations at t= 0.
Oth-erwise, the only consistent perturbative solution would be the trivial solution with δφ= 0 and/or δHi= 0.
Note that an unstable mode with p= J1H0indicates that
the perturbative solution will remain stable for a brief period of time of the order of 1/(J1H0). This means that the de
Sitter solution will not be stable once t > 1/(J1H0). The
exact decaying process will, however, also depend on the dynamics of the scalar field. For a slow roll-over scalar field, the system may remain close to the de Sitter phase for a brief period of time in competition with the instability period
t∼ 1/(J1H0)derived from the unstable mode p= J1H0.
4 Higher derivative induced gravity model
For a simple demonstration in this section, we will focus on the higher derivative induced gravity model given by
L= − 2φ 2R− αR2− βRμ νRμν + γ φ2R μν βγR βγ σρRμνσρ− 1 2∂μφ∂ μφ− V (φ) ≡ 2φ 2L 1+ L2+ γ φ2L3+ Lφ (28) with L1= −R, L2= −αR2− βRμνRμν, L3= RβγμνRσρβγRμνσρ and Lφ = −12∂μφ∂μφ− V (φ) denoting the lowest order curvature coupling, the higher order terms, and the scalar field Lagrangian, respectively. By definition the induced gravity models assume that all dimensionful parameters and all coupling constants, except the symmetry breaking scale parameter φ0, are induced by some proper choices of the
dynamical fields. For example, the gravitational constant is replaced by 8π G= 2/( φ2) as a dynamical field. In addition, the cosmological constant becomes V (φ) in this model. There is no need for any induced parameters for the quadratic terms R2and R2μνbecause the coupling constants
αand β are both dimensionless by themselves. The action of this system is also invariant under the global scale trans-formation gμν→ Λ−2gμνand φ→ Λφ with some arbitrary constant parameter Λ.
The corresponding Lagrangian can be shown to be
L= φ2(2A+ B + 2C + D)
− 4α 4A2+ B2+ 4C2+ D2+ 4AB + 8AC + 4AD
+ 4BC + 2BD + 4CD − 2β 3A2+ B2+ 3C2+ D2
+ 2AB + 2AC + 2AD + 2BC + 2CD
+ 8γ φ2 2A3+ B3+ 2C3+ D3 +1 2 ˙φ 2− V (φ) (29)
in the Kantowski–Sachs type spaces. Here A= ˙H1+ H12,
B = H12 + k/a2, C = H1Hz, D= ˙Hz+ Hz2. This La-grangian can be shown to reproduce the de Sitter models when we set Hi→ H0in the isotropic limit.
The Friedmann equation (10) reads 1 2 φ 2DL 1+ DL2+ γ φ2DL3+ φ ˙φHiL i 1− 2 γ φ3˙φHiL i 3 =1 2 ˙φ 2+ V (φ) (30)
for the induced gravity model. In addition, the scalar field equation (11) can be shown to be:
¨φ + 3H0˙φ + V= φL1− 2γ
φ3L3 (31)
As a result, the leading order Friedmann equation and the scalar field equation can be shown to be
V0≡ V (φ0)= 3 0φ02H02 (32)
V(φ0)= 12 0φ0H02 (33)
in the presence of the de Sitter solution with φ= φ0 and
The conventional approach assumes that the scalar field is a slow roll-over field obeying ¨φ V and H0˙φ V near
the inflationary phase. It can be shown that in the de Sit-ter inflationary phase, the dynamical part of the scalar field equation evolves as ¨φ+ 3H0˙φ ∼ 0. This equation leads to
the approximate solution
φ∼ φ0+ ˙φ0
3H0
1− exp(−3H0t ) (34)
This result is clearly consistent with the slow roll-over as-sumption we just made. In summary, the zeroth order equa-tions lead to a few constraints on the field parameters:
4V0= φ0
∂V
∂φ(φ= φ0)= 12 0φ
2
0H02 (35)
An appropriate effective spontaneously symmetry breaking potential V of the form
V (φ)=λ
4
φ2− φ022+ 6 0H02φ2− φ20+ 3 0H02φ02 (36) with arbitrary coupling constant λ, can be shown to be a good candidate satisfying all the scaling conditions (35).
The value of H0can be chosen to induce enough inflation
for a brief moment as long as the slow roll-over scalar field remains close to the initial state φ= φ0. The de Sitter phase will hence remain valid and drive the inflationary process for a brief moment governed by the decaying speed of the scalar field.
The inflationary Hubble parameter H0is related to γ , 0,
and V0by the following equation:
H06− φ 2 0 8γ H 2 0+ V0 3 φ02= 0 (37)
This equation can be solved to give
H02= φ02 6γ cos θ0∓ π 3 (38)
with cos θ0≡√6γ / V0/[ φ03]. As a result,
0= 1− 4 3φ02cos 2 θ0∓ π 3 (39)
A different choice of 0is therefore equivalent to a different
choice of initial state V0and vice versa. For the practical
rea-sons, we can take either 0or V0related by above equation
as a free parameter.
Note that the local extremum of this effective potential can be shown to be φ= 0 (local maximum) and φ2= φm2 =
φ02− 12 0H02/λ < φ02(local minimum). In addition the min-imum value of the effective potential can be shown to be
Vm= V0− 36 02H04/λ < V0 (40)
The constraint Vm>0 implies that λφ02 >12 0H02. Or
equivalently, it implies that φm2 >0. In addition, we will set
φm2/2= 1/(8πG) = 1 in Planck units for convenience in this paper.
When the scalar field settles down to the local minimum
φm of the effective potential at large time in the post infla-tionary era, it will oscillate around the local minimum and kick off the reheating process. The scalar field will eventu-ally become a constant background field and induces a small cosmological constant Vm= V (φm).
The final state φ= φmrequires the identity
0H02= mHm2 (41) for the consistency of a stable final state. Here m≡ [1 − 8γ Hm4/( φm4)]. By solving Hm2 as a function of H02, we can
obtain Hm2= φ4 m 6γ cos θm∓ π 3 (42)
with the following constraint:
cos θm= 54γ φ4 m 1/2 1−8γ H 4 0 φ40 ≤ 1 (43) Hence we have 0 ≤ φm4 54γ 1/2 = 2 27 γ 1/2 (44)
This implies the inequality (27 02− 16H04/φ40)γ ≤ 2 0. Therefore, we have
γ≤ 2 0
27 02− 16H04/φ40 (45)
if the denominator 27 02− 16H04/φ04is positive. Otherwise, the inequality (27 20− 16H04/φ40)γ ≤ 2 0is automatically satisfied. In addition, the leading order perturbation equation in δH and δφ of the Friedmann equation in this model can be shown to be 4 3α+ β − 6γH 2 0 φ02 δ ¨H+ 3H0δ ˙H+ KH02δH = 1− 24γ H 4 0 φ04 φ0[δ ˙φ − H0δφ] (46) with K= 24γ H 4 0/φ 2 0− φ 2 0 4(3α+ β − 6γ H02/φ02)H02
Similarly, the leading perturbation equation of the scalar field can be shown to be
with J= V− 12 0H02− 384γH 6 0 φ04 H0−2
The variational equation of azcan be shown explicitly to be redundant in the limit Hi= H0+ δHi and φ= φ0+ δφ following the Bianchi identity.
Assuming that δH = exp[hH0t]δH0 and δφ =
exp[pH0t]δφ0 for some constants h and p, we can write the above equations as
1− 24γ H 4 0 φ40 φ0[p − 1]δφ = 4 3α+ β − 6γH 2 0 φ02 H0 h2+ 3h + K δH (48) H0 p2+ 3p + J δφ= 6 0φ0[h + 4]δH (49)
These equations are consistent when all coefficients vanish simultaneously. This implies that h= −4 and p = 1. This set of solution (h, p)= (−4, 1) hence imposes two addi-tional constraints: − 16(3α + β)H 2 0 φ02 + 72γ H04 φ04 = 0− 16(3α + β)H02 φ20 + 80γ H04 φ04 = 0 (50) λ= 192γH 6 0 φ06 − 2 H02 φ20 (51) with 2λφ02= V0− 12 0H02.
The coupling constant λ has to positive in order for the effective potential V (φ) to be free from run-away negative global minimum at φ → ∞. As a result, the constraints
0>0 and λ > 0 imply that
φ04
96H04< γ <
(3α+ β)φ02
5H02 (52)
Together with the constraint (45),
γ≤ 2 0
27 02− 16H04/φ04
the physical parameters such as γ can be chosen properly to accommodate a large class of solutions to the evolu-tion of our physical universe. As a result, the inflaevolu-tionary phase will remain stable against small perturbation along the
δH (= exp[−4H0t]δH0) direction. On the other hand, the inflationary phase also has an unstable mode when we per-turb the system along the δφ(= exp[H0t]δφ0)direction that will hold the de Sitter phase stable only for a brief moment
t∼ 1/(pH0)= 1/H0. This brief period is apparently not enough for a complete inflationary phase. As indicated from (34), φ does not evolve appreciably during the inflationary phase if the unstable mode will not break the stability of the system. In short, enough inflation will require an unstable mode with a long enough t before the exit of the inflation-ary phase.
In addition to the above trivial solution, there are some other perturbation solutions. Note that the perturbation equations can also be cast in the form
DδΨ = D δH δφ = A1(h2+ 3h + K) −C1(h− 1) B1(h+ 4) −(h2+ 3h + J ) δH δφ = 0 (53)
with δH ≡ kHexp[hH0t], δφ ≡ kφexp[hH0t]. Here we have assumed that δH = ikiexp[hiH0t] and δH =
ijiexp[hiH0t] such that
DδΨ= i D ki ji exp[hiH0t] = 0 (54)
Hence solving the perturbation equations amounts to solv-ing the eigenvalue problem given by (53). It is also un-derstood that h written in (53) represents the eigenvalue
h of the operator ∂t operating on its eigenstate, namely,
∂texp[hH0t] = hH0exp[hH0t]. In order to simplify the derivation, we will extract all dimensionful parameters by defining ϕ0= φ02/H02, β1= (3α + β)/ϕ0, γ1= γ /ϕ0−2and
λ1= λϕ0. As a result, the coefficients A1, B1, C1, J and K
can be written as A1= 4(β1− 6γ1)ϕ0 (55) B1= 6( − 8γ1)√ϕ0 (56) C1= ( − 24γ1)√ϕ0 (57) J= 2λ1− 384γ1 (58) K= 24γ1− 4(β1− 6γ1) (59)
The perturbation equations have a non-trivial solution only when det D= 0, which can be written as
2+ F + G = 0 (60) with ≡ h2+ 3h and F=(24γ1− )(1 + 6 − 48γ1) 4(β1− 6γ1) + 2λ1− 384γ1 (61) G=(24γ1− )(λ1− 12 − 96γ1) 2(β1− 6γ1) (62)
Therefore, we can solve the perturbation equations and ob-tain the eigenvalue h as
h=−3 ± √ 9+ 4 2 (63) with =−F ± √ F2− 4G 2 (64)
Hence we find four independent solutions to the perturba-tion equaperturba-tions (53). The graceful exit requires the exist of at least an unstable mode with h > 0. This will be the case if
>0. In fact, an inflationary phase for a period t ∼ 60/H0is required for the universe to undergo enough
expan-sion of roughly exp[60] times before the end of the inflation-ary phase. This in turns requires that h∼ 1/60. It is easy to show that this condition is equivalent to the constraint ∼ 1/5. This condition can be shown to be F + 5G ∼ −1/5. Hence it can also be written explicitly as
λ1∼ 4(β1− 6γ1)(1920γ1− 1) + 5(24γ1− ) ×(1080γ1+ 114 − 1)/10(96γ1− 5 + 4β1) (65) as a constraint on λ1. Note that in the limit γ1= 0, we have
λ1∼
5 (1− 114 ) − 4β1 10(4β1− 5 )
(66)
Together with the constraint (45), which is equivalent to
2ϕ02≥ 27γ1 − 16γ1+ 64γ 2 1 (67)
the unstable mode can be managed to provide reasonable resolution to the graceful exit problem for the inflationary models. For example, we can choose
γ1> 1− 114 1080 (68) and either β1 6 > γ1> 24> 1 1920 (69) or β1 6 > γ1> 1 1920> 24 (70)
as the constraint on the coupling constants , β1, and γ1to
ensure that λ1>0. Note that the inequality β1/6 > γ1>
/24 can be shown to imply that 96γ1− 5 + 4β1>0. We
can also show that (69) implies that > 1/80. As a result, the inequality (68) implies that
γ1>−
17 43200 >
1− 114 1080
On the other hand, the inequality (70) implies that < 1/80. As a result, the inequality (68) implies that
γ1>
1− 114 1080 >−
17 43200
Therefore, we can indeed choose proper constraints on the coupling constants to ensure the resolution of graceful exit problem in the inflationary universe. Both constraints shown above can be realized with reasonably chosen coupling con-stants. Therefore, the scalar field does provide a useful tool both in inducing proper inflation and providing a natural mechanism for the graceful exit problem.
Note again that when the scalar field settles down to the local minimum φm of the effective potential at large time in the post-inflationary era, it will oscillate around the local minimum and kick off the reheating process. The scalar field will eventually become a constant background field with a small cosmological constant Vm= V (φm). The stability of the system will then be dominated by the evolution of the scale factor a. Therefore, we end up with a stable de Sitter background space in the large time region.
The reason that only the special combination 3α+ β shows up in the stability equation is that two identities connect the quadratic curvature terms in the 4-dimensional space time. Indeed, there are a Gauss–Bonnet invariant E and an additional conformal Weyl invariant at our disposal:
E= RabcdRcdab− 4RbaRab+ R2 (71)
C2≡ CabcdCabcd= RcdabRabcd− 2RbaRab+1
3R
2
(72)
As a result, we can write
3αR2+ 3βRabRba= (3α + β)R2+3β
2
C2− E (73) The Gauss–Bonnet term√gEis an Euler invariant known to be a total derivative. Therefore, it will not contribute to the field equation. In addition, the FRW space is known to be conformally flat. Hence the conformal Weyl invariant C2 will not contribute to the field equations either. Therefore, the stability equation will depend only on the combination 3α+ β.
The reason that the quadratic terms do not affect the scale of inflation H0can be checked readily by showing that any
quadratic Lagrangian of the combinations l1H˙2+l2( ˙H H2+
H4)will not contribute to the Friedmann equation. Here li are constants. Both R2 and RbaRba are of this form, hence they will not contribute to the background Friedmann equa-tion. Note that the curvature term is assumed to be negligible in this phase. Alternatively, we can focus on the flat homo-geneous space for simplicity.
Indeed, the quadratic terms will contribute to the Fried-mann equation as a combination of
E2 = L + Hi d dt + 3H Li− HiLi− ˙HiLi → L + 3H2L ˙ H− HLH (74)
in the de Sitter background with LH ≡ δL/δH and LH˙ ≡
δL/δ ˙H. It is clear that the l1term vanishes in the de Sitter
space. Furthermore, ˙H H2 terms will not contribute to the
E2except through the effect of LH˙. Hence H LH → 4L in the de Sitter space for the quadratic Lagrangian. As a result
E2→ 3(H2LH˙ − L). Therefore, E2= 0 if and only if the contributions of ˙H H2and H4in the quadratic Lagrangian are equal as stated in the form l2( ˙H H2+ H4)shown above.
In the de Sitter background space, the Riemannian curva-ture component functions A, B, C, D for the KS type space defined earlier are related to each other by A= D and
B= C when the curvature term is negligible in the
inflation-ary phase. Therefore any combinations of the forms A2+B2 and AB all fall into the class of l2( ˙H H2+ H4). Therefore
it is straightforward to verify that the quadratic Lagrangian does not contribute to the Friedmann equation in the de Sit-ter background.
5 Einstein gravity and induced gravity
In order to compare and clarify the differences of the stabil-ity conditions contributed from the higher derivative terms and the induced gravity models with respect to the Einstein gravity, we will also study the stability conditions of the Ein-stein theory and induced gravity model without higher deriv-ative terms in this section.
5.1 Leading order induced gravity model
For the induced gravity model, the Lagrangian of the system is L= − 2φ 2R−1 2∂μφ∂ μφ− V (φ) (75)
The Friedmann equation (10) reads 1 2 φ 2DL 1+ φ ˙φHiLi1= 1 2 ˙φ 2+ V (φ) (76)
for the induced gravity model (75) with L1= −R. In
addi-tion, the scalar field equation (11) can be shown to be:
¨φ + 3H ˙φ + V= − φR (77)
As a result, we also end up with the constraint (35),
4V0= φ0
∂V
∂φ(φ= φ0)= 12 0φ
2 0H02
in the presence of the de Sitter background solution with
φ= φ0and Hi= H0for all directions.
In addition, the slow roll-over field obeys ¨φ V and
H0˙φ Vnear the inflationary phase. It can be shown that
in the de Sitter inflationary phase, the dynamical part of the scalar field equation also evolves as ¨φ+ 3H0˙φ ∼ 0. This
equation leads to the approximate solution (34)
φ∼ φ0+ ˙φ0
3H0
1− exp(−3H0t )
This result is clearly consistent with the slow roll-over as-sumption we just made. An appropriate effective sponta-neously symmetry breaking potential V is therefore the same as the one given in (36):
V (φ)=λ
4
φ2− φ202+ 6 0H02φ2− φ02+ 3 0H02φ02
with arbitrary coupling constant λ. As a result, the stability conditions for the models without higher derivative terms will therefore be the α= β = γ = 0 limit of the higher derivative models discussed in Sect.4. For example, the sta-bility equations become
− φ2
0δH= φ0[δ ˙φ − H0δφ] (78)
δ ¨φ+ 3H0δ ˙φ+ J0H02δφ= 6 0φ0(δ ˙H+ 4H0δH ) (79)
with J0 = (V− 12 H02)H0−2. Similarly, the variational
equation of azcan be shown explicitly to be redundant in the limit Hi= H0+δHiand φ= φ0+δφ following the Bianchi identity. In this case, the trivial solution (h, p)= (−4, 1) will not survive for the independent perturbations δH =
kHexp[hH0t] and δφ = kφexp[pH0t]. Instead, there is a consistent solution of the form with δH ≡ kHexp[hH0t],
δφ≡ kφexp[hH0t]. Note that the perturbation equations can also be cast in the form
DδΨ = D δH δφ = φ0/H0 (h− 1) 6 φ0(h+ 4) −H0(h2+ 3h + J0) δH δφ = 0 (80)
Note that non-trivial solutions to the above equation exist only when det D= 0. Therefore, we can derive the follow-ing stability equation for h:
h2+ 3h + K0= 0 (81)
with K0= [2λφ02/H02− 24 ]/[1 + 6 ]. As a result, the
so-lution h to above stability equation can be shown to be
h= h±=1 2 −3 ± 9− 8λφ 2 0− 24 H02 (1+ 6 )H02 1/2 (82)
Consequently, an unstable mode, h+ >0, exists when
λφ02<24 H02. Therefore, the unstable mode can provide a natural way to end the inflationary phase. Note that the sta-bility equation (81) is a polynomial equations of degree 2, a simplified version of the degree 4 polynomial equation (60) for the higher derivative models. It is therefore straight-forward to observe the critical role of the higher derivative terms by comparing these two equations.
5.2 Einstein theory with a scalar field
Let us consider further the Einstein theory with a coupled scalar field in the absence of the higher derivative terms:
L= −1
2R− 1 2∂μφ∂
μφ− V (φ) (83)
The Friedmann equation reads
H12+ 2H1Hz= 1 2 ˙φ
2+ V (φ) (84)
for the induced gravity model (83). Similarly, we have ig-nored the curvature term during the inflationary era. In ad-dition, the scalar field equation can be shown to be
¨φ + 3H0˙φ + V= 0 (85)
As a result, we also end up with the leading order Friedmann equation and the scalar field equation as
V0= 3H02 (86)
V(φ0)= 0 (87)
in the presence of the de Sitter solution with φ= φ0 and
Hi = H0 for all directions. The perturbation equations are
δH= 0 and
p2+ 3p + V0/H02δφ= 0
for δφ∼ exp[pH0t]. Note that h-mode and p-mode
decou-ple in this set of equations. Therefore, there are only trivial solutions for these models. Indeed, the h-mode equation im-plies that the h-mode is a stable mode. Therefore, the scalar
p-mode will have to take care of the graceful exit mecha-nism in these models. For example, the model with a sym-metry breaking scalar potential
V = λφ2− v22/4 (88) will ensure the constraints (86) and (87) remain consistent if the evolution starts with φ0= 0. Here v is a constant
de-noting the symmetry breaking scale given by the relation
λv4/4= 3H02. Indeed, the constrain V(φ0)= 0 indicates
that the scalar field in the inflationary era has to start off from the local maximum, φ= 0, of the scalar potential and
rolls slowly down toward the local minimum, φ= v, of the scalar potential. Hence the solution of the δφ equation is
p= p±=1
2
−3 ± 9+ 4λv2 1/2 (89)
As a result, p+>0 is an unstable mode. Hence the grace-ful exit can occur if the evolution starts from a field con-figuration H0and φ0= 0 under the effect of the symmetry
breaking potential (88).
Einstein–Hilbert model
If we turn off the scalar field in (83), the system will become the Einstein–Hilbert model with a cosmological constant Λ:
L= −1
2R− Λ (90)
The situation is similar to the model with scalar field. The Friedmann equation reads
H12+ 2H1Hz= Λ (91)
for the pure gravity model (90). Therefore, the system will remain stable for a long time by itself. This is also the reason why a scalar field or the higher derivative term is needed for the graceful exit.
6 Conclusion
The existence of a stable de Sitter background is closely re-lated to the choices of the coupling constants in the system. The pure higher derivative gravity model with the quadratic terms and a cubic interaction is known to admit a stable in-flationary solution with a proper choice of the field para-meters [31,32]. Indeed, proper choice of the coupling con-stants enables the existence of a de Sitter phase that is sta-ble against any small isotropic and anisotropic perturbations. In many cases, there also exists another unstable mode that will be acting in favor of the graceful exit of the inflationary models.
It is shown that any small perturbation (against the isotropic FRW background space) and any small pertur-bation (against the anisotropic KS type background space) obey the same perturbation equations. This is also true for modified gravity models. Therefore, the stable modes will act in favor of the stability of the background de Sitter space. These stable modes will also ensure that the anisotropy of the de Sitter space will not grow out of control. On the other hand, the unstable mode indicates that the isotropic back-ground is unstable against any small isotropic or anisotropic perturbations. Therefore, only a small anisotropy in the early universe could be generated by arbitrary small anisotropic
perturbations in both of these models. Hence we are look-ing for constraints on the field parameters that will ensure that the system admits at least an unstable mode for the resolution of the graceful exit problem for the inflationary solution.
Indeed, we have shown that various constraints must be observed for the existence of an unstable mode in the modi-fied gravity models. In particular, we show that, for induced gravity models, an unstable mode does exist with prop-erly chosen constraints. As a result, only a small anisotropy against the de Sitter background can grow during the infla-tionary phase for this induced model. Indeed, an explicit model with a spontaneously symmetry breaking φ4 poten-tial is presented as an example for a simple demonstration. Accordingly, various constraints are also derived for this model. In addition, we also compare the higher derivative models with the models without higher derivative terms. The differences with respect to the Einstein gravity are also clar-ified in previous section. As a result, the effect of the higher derivative terms become more transparent by these compar-isons.
In summary, we have shown that an unstable mode for a small (an)isotropic perturbation against the de Sitter back-ground does exist for the induced gravity model The prob-lem of a graceful exit can be achieved counting on the unsta-ble mode of the scalar field perturbation. In addition, we also explain explicitly the reason that the quadratic terms will not affect the inflationary solution characterized by the Hubble parameter H0. These quadratic terms play, however, a
criti-cal role in the stability problem of the de Sitter background in the modified gravity models.
Acknowledgements This work is supported in part by the National Science Council of Taiwan. The author would like to thank the referee for useful suggestions and comments.
References
1. S. Gulkis, P.M. Lubin, S.S. Meyer, R.F. Silverberg, Sci. Am.
262(1), 122–129 (1990)
2. D.J. Fixsen et al., Astrophys. J. 473, 576–587 (1996)
3. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)
4. C.W. Misner, K. Thorne, T.A. Wheeler, Gravitation (Freeman, San Francisco, 1973)
5. R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984)
6. E.W. Kolb, M.S. Turner, The Early Universe (Addison-Wesley, Reading, 1990)
7. A.H. Guth, Phys. Rev. D 23, 347 (1981)
8. E.W. Kolb, M.S. Turner, Ann. Rev. Nucl. Part. Sci. 33, 645 (1983) 9. F.S. Accetta, D.J. Zoller, M.S. Turner, Phys Rev. D 31, 3046
(1985)
10. W.F. Kao, Class. Quantum Gravity 24, 4295 (2007). hep-th/ 0605078
11. M.B. Green, J.H. Schwartz, E. Witten, Superstring Theory (Cam-bridge University Press, Cam(Cam-bridge, 1986)
12. A.S. Goncharov, A.D. Linde, V.F. Mukhanov, Int. J. Mod. Phys. A 2, 561 (1987)
13. N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982)
14. G.F. Chapline, N.S. Manton, Phys. Lett. B 120, 105 (1983) 15. C.G. Callan, D. Friedan, E.J. Martinec, M.J. Perry, Nucl. Phys. B
262, 593 (1985).gr-qc/0102077
16. A. Dobado, A. Lopez, Phys. Lett. B 316, 250 (1993)
17. A. Dobado, A.L. Maroto, Phys. Rev. D 52, 1895–1901 (1995). Erratum: ibid. D 53 (1996) 2262.hep-ph/9406409
18. P. Kanti, J. Rizos, K. Tamvakis, Phys. Rev. D 59, 083512 (1999) 19. W.F. Kao, U.-L. Pen, Phys. Rev. D 44, 3974–3977 (1991) 20. W.F. Kao, U.-L. Pen, P. Zhang, Phys. Rev. D 63, 127301 (2001).
gr-qc/9911116
21. K.I. Maeda, Phys. Rev. D 39, 3159 (1989) 22. R. Wald, Phys. Rev. D 28, 2118 (1983)
23. J.D. Barrow, S. Hervik, Phys. Rev. D 73, 023007 (2006) 24. J.D. Barrow, S. Hervik, Phys. Rev. D 74, 124017 (2006) 25. C.-M. Chen, T. Harko, M.K. Mak, Phys. Rev. D 63, 064002
(2001).hep-th/0005236
26. C.-M. Chen, W.F. Kao, Phys. Rev. D 64, 124019 (2001). hep-th/0104101
27. S.A. Abel, G. Servant, Nucl. Phys. B 597, 3–22 (2001).hep-th/ 0009089
28. X.-z. Li, J.-g. Hao, Phys. Rev. D 68, 083512 (2003).astro-ph/ 0308288
29. S. Shu, NCTU, Master thesis
30. S. Nojiri, O. Obregon, S.D. Odintsov, K.E. Osetrin, Phys. Rev. D
60, 00 (1999).hep-th/9902035
31. W.F. Kao, Phys. Rev. D 74, 043522 (2006) 32. W.F. Kao, Eur. Phys. J. C 53, 87 (2008)
